The final answer to the complexity of a basic problem in resilient network design

Tomaszewski, Artur

doi:10.1016/j.endm.2013.05.125

Cited by 4 publications

(5 citation statements)

References 6 publications

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“…For instance let us consider RR: taking a = 0 implies that flow thinning is not allowed, while b → ∞ means that new rerouting paths can be created in practice; this is because the flow on some paths can be enlarged at any finite value starting from practically insignificant flow values. All these special cases have different levels of complexity for the single link failure case: GR and PD fall into the polynomial time complexity class [2], while RR is shown to be N P-hard for both the directed [9,18] and undirected [20] cases. The observation suggests that both problems will exhibit the same N P complexity.…”

Section: Complexity Discussionmentioning

confidence: 99%

Generalized elastic flow rerouting scheme

et al. 2015

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The present study deals with Elastic Flow Rerouting (EFR)-an original traffic restoration strategy for protecting traffic flows in communication networks (including wireless networks) against multiple link failures. EFR aims at alleviating the trade-off between practicability of traffic restoration and the cost of network resources observed in existing networking solutions. We present an extension of EFR capable of managing multiple partial link failures. We describe EFR and its extension, formulate the EFR related optimization problems, and discuss approaches for their resolution. We also discuss numerical results illustrating effectiveness of EFR in terms of the link capacity cost.

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Section: Complexity Discussionmentioning

confidence: 99%

Generalized elastic flow rerouting scheme

et al. 2015

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“…When the number of failing links is not bounded by a constant k and grows with the size of the graph, as for example, for the full single links failure scenario (recall that this means that all links in \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{E}\end{align*} \end{document} can fail, one at a time), then the approach to FR described above becomes exponential. But this could be expected as we know from a recent result of [22] that FR becomes \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{NP}\end{align*} \end{document} ‐hard (already for one demand) when the number of failing links in a single links failure scenario is not bounded by a constant.…”

Section: Complexity Of Variants Of Frmentioning

confidence: 99%

“…So in acyclic graphs, RFR is also polynomial for a bounded number of the single link failures. But RFR is \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{NP}\end{align*} \end{document} ‐hard when the number of failing links in a single links failure scenario is not bounded because the proof of [22] works for both acyclic and nonacyclic graphs.…”

Section: Complexity Of Variants Of Frmentioning

confidence: 99%

“…Their proof, based on reduction from a Hamiltonian chain problem, is valid for also for circuit‐free paths and thus for RFR, although this is not explicitly stated in the article. Still, the complexity of FR (and of RFR) in this case is open since the proof in [22] is valid only for directed graphs.…”

Section: Complexity Of Variants Of Frmentioning

confidence: 99%

“…1 1 This has been proved only recently by a co‐author of this article in [22] so this fact is not widely known. …”

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confidence: 98%

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Complexity of a classical flow restoration problem

et al. 2013

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In this article, we revisit a classical optimization problem occurring in designing survivable multicommodity flow networks. The problem, referred to as FR, assumes flow restoration that takes advantage of the so-called stub release. As no compact linear programming (LP) formulation of FR is known and at the same time all known noncompact LP formulations of FR exhibit N Phard dual separation, the problem itself is believed to be N P-hard, although without a proof. In this article, we study a restriction of FR (RFR) that assumes only elementary (cycle-free) admissible paths-an important case virtually not considered in the literature. The two problems have the same noncompact LP formulations as they differ only in the definition of admissible paths: all paths (also those including cycles) are allowed in FR, while only elementary paths are allowed in RFR. Because of that, RFR is in general computationally more complex than FR. The purpose of this article, is three-fold. First, the article reveals an interesting special case of RFR-the case with only one failing link-for which a natural noncompact LP formulation obtained by reducing the general RFR formulation still exhibits N P-hard dual separation, but nevertheless this special case of RFR is polynomial. The constructed example of a polynomial multicommodity flow problem with difficult dual separation is of interest since, to our knowledge, no example of this kind has been known. In this article, we also examine a second special case of RFR, this time assuming two failing links instead of one, which turns out to be N P-hard. This implies that problem RFR is N P-hard in general (more precisely, for two or more failure states). This new result is the second contribution of the article. Finally, we discuss the complexity of FR in the light of our new findings, emphasizing the differences between RFR and FR.

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